Question

The wavelength of a probe is roughly a measure of the size of a structure that it can probe in some detail. The quark structure of protons and neutrons appears at the minute length-scale of 10–15 m or less. This structure was first probed in early 1970’s using high energy electron beams produced by a linear accelerator at Stanford, USA. Guess what might have been the order of energy of these electron beams. (Rest mass energy of electron = 0.511 MeV.)

Answer 1

Given: The wavelength of a proton or a neutron is 10 −15  m and the rest mass energy of electron is 0.511 MeV.

de Broglie wavelength is given as,

λ= h p

Where, p is the momentum and h is the Planck’s constant.

By substituting the given values in the above expression, we get

10 −15 = 6.6× 10 −34 p p=6.6× 10 −19   kgms -1

The relativistic kinetic energy is given as,

E 2 = p 2 c 2 + m e 2 c 4 = ( pc ) 2 + ( m e c 2 ) 2

Where, E is the kinetic energy, c is the speed of light and m e is the rest mass of an electron.

By substituting the values in the above equation, we get

E 2 = ( 6.6× 10 −34 ×3× 10 8 ) 2 + ( 0.511× 10 6 ×1.6× 10 −19 ) 2 =392.04× 10 −22 E= 392.04× 10 −22 =1.98× 10 −10  J

Further simplifying the above expression, we get

E=1.98× 10 −10  J× 1 eV 1 .6×10 -19  J =1.24× 10 9  eV

Thus, the electron energy emitted from the accelerator at Stanford, USA will be of the order 1.24× 10 9  eV.